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Measuring the impact of observations on the predictability of the

Kuroshio Extension in a shallow-water model

Werner Kramer∗ and Henk A. Dijkstra

Institute for Marine and Atmospheric Research Utrecht, Department of Physics and Astronomy

Utrecht University, Utrecht, Netherlands

Stefano Pierini

Dipartimento di Scienze per l’Ambiente, Universita di Napoli Parthenope, Naples, Italy

Peter Jan van Leeuwen

Department of Meteorology, University of Reading, Reading, United Kingdom

∗Corresponding author address: Institute for Marine and Atmospheric Research Utrecht, Dept. of Physics

and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands.

E-mail: w.kramer@uu.nl

1

ABSTRACT

In this paper sequential importance sampling is used to assess the impact of observations on

a ensemble prediction for the decadal path transitions of the Kuroshio Extension (KE). This

particle filtering approach gives access to the probability density of the state vector, which

allows us to determine the predictive power — an entropy based measure — of the ensemble

prediction. The proposed set-up makes use of an ensemble that, at each time, samples

the climatological probability distribution. Then, in a post-processing step, the impact of

different sets of observations is measured by the increase in predictive power of the ensemble

over the climatological signal during one-year. The method is applied in an identical-twin

experiment for the Kuroshio Extension using a reduced-gravity shallow water model. We

investigate the impact of assimilating velocity observations from different locations during

the elongated and the contracted meandering state of the KE. Optimal observations location

correspond to regions with strong potential vorticity gradients. For the elongated state the

optimal location is in the first meander of the KE. During the contracted state of the KE it

is located south of Japan, where the Kuroshio separates from the coast.

1. Introduction

Our view of the oceans and atmosphere comes from remote sensing measurements and

from still relatively sparse observations. Data assimilation combines the information gained

from observations with computer simulations to obtain a three-dimensional representation

of the current state of the ocean and atmosphere. An accurate estimate of the current state

is particularly important for forecasting the future evolution. Due to the chaotic nature of

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these dynamical systems, uncertainties present in the initial state grow rapidly reducing the

utility of the forecast. In this paper we describe an ensemble-based method for measuring the

effect of observations. Using model-generated sea-surface height observations, we apply the

method to determine the best location for monitoring the decadal variability of the Kuroshio

Extension (KE).

The Kuroshio Extension is the eastward flowing free jet formed when the Kuroshio current

detaches from the Japanese coastline. The KE region has the largest sea-surface height

(SSH) variability on sub-annual and decadal time scales in the extratropical North Pacific

Ocean (Qiu 2002). The decadal variability is related to transitions of the KE from a highly

energetic elongated path to a weaker contracted and more convoluted path (Qiu and Chen

2005). Prediction of the path of the Kuroshio is very important for local fisheries and hence

local economies (Kagimoto et al. 2008). The position of the Kuroshio strongly determines

the regions where phytoplankton and hence fish is located. The KE variability has been the

subject of a number of experimental studies, like WESTPAC at 152E (Schmitz Jr. et al.

1982), the Kuroshio Extension Region Experiment (KERE) at 143E (Hallock and Teague

1995) and the Kuroshio Extension System Study (KESS) at 146E (Donohue et al. 2008).

Pierini (2006) obtained a reasonably successful comparison of satellite SSH observations

and the results from a reduced-gravity shallow-water model of the western Pacific Ocean.

The model is forced by a steady wind-stress forcing and has only one active layer. In this

model, not only the decadal transitions between contracted and elongated Kuroshio paths

are found (Pierini et al. 2009), but the migration of the northward extension of the Kuroshio

correspond well with those determined from observations (Qiu and Chen 2005). Important

to note is that the model also captures the high-frequency variability during the northward

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migration of the KE. The observed temporal variability is, however, higher than the modeled

one and on smaller time scales. This is not surprising as the model is only eddy permitting

and does not capture baroclinic instability.

The physics behind the bimodality of the KE are not well understood and still open

for discussion. Pierini et al. (2009) explain the two states of the KE, its decadal period

and the changing high-frequency variability in terms of nonlinear dynamical systems theory.

The success of the reduced-gravity shallow-water model suggests that the internal ocean

mechanics are responsible. Another explanation is sought in the variable atmospheric forcing

(Miller et al. 1998; Deser et al. 1999; Qiu 2003; Qiu and Chen 2005). Here, westward

propagating SSH anomalies, generated in the eastern North Pacific by wind stress anomalies

at different phases of the Pacific decadal oscillation, cause the positional shifts of the KE

jet.

Objective measures for the impact of observations on analysis uncertainties are impor-

tant for both the design of persistent observation networks (e.g. mooring arrays) and the

deployment of supplementary or targeted observations (e.g. airplane reconnaissance and

dropsondes). Targeted observations are used to decrease the forecast uncertainty of high

risk events, like hurricanes. For an overview on targeted observations procedures and is-

sues see Langland (2005). An objective procedure, based on e.g. singular vectors, is used

to determine regions with fast-growing initial errors. A targeted observation is considered,

when the forecast error can be decreased by assimilating additional observation data. Baker

and Daley (2000) argued that none of the traditional techniques consider the characteristics

of the data-assimilation systems used. As such, interaction with the background field of

the analysis and interactions with other observations are ignored. For this reason objective

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procedures for adaptive observations are developed that incorporate the data-assimilation

techniques (e.g. the ensemble transform Kalman filter, Bishop et al. (2001)).

Objective procedures for targeted observations are not necessarily good for optimizing

array designs. These methods aim to improve the forecast at a given time, while mooring

arrays are operated over longer times. Sakov and Oke (2007) used an ensemble of sys-

tem states from a long model run or from observation based gridded fields to represent the

time-averaged statistics of the system. The system states ensemble is used to obtain the

background error covariances, which describe the uncertainty in the model state when no

observational data is available. The optimal set of observations is then obtained by minimiz-

ing the analysis error covariance using Kalman filter theory. Problems arise when the time

averaged-statistics are strongly non Gaussian, which makes an analysis based on covariances

meaningless. As such, the bimodality of the Kuroshio Extension will present an obstacle for

this method.

In this paper we want to answer the question what is the best measurement location to

monitor the Kuroshio Extension system for optimal prediction of its decadal transitions. The

general idea is to perform an ensemble model forecast, which in our case is obtained with the

reduced-gravity shallow-water model of Pierini (1996, 2006). The ensemble samples the time-

averaged probability distribution of the system state. Observations are then used to update

the ensemble using a particle filtering technique. For this approach, neither the assumption

of Gaussian error statistics, nor the linearization of the model are required. The general

methodology is described in section 2. The set-up allows us to find which measurements

are most successful in increasing the predictive skill or power of the ensemble forecast. In

section 3 results of an identical-twin experiment for the KE region are presented. We use

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this experiment to investigate the assets and drawbacks of the methodology. The test is

done by assimilating velocity observations at four preselected locations. Thereafter, we use

the method to find the optimal location for following the decadal transition in the KE. Final

comments and conclusions are made in section 5.

2. Methodology

a. Reduced-gravity shallow-water model

In this paper we utilize the reduced-gravity shallow-water model and set-up used by

Pierini (2006) to investigate the decadal oscillations of the Kuroshio Extension. The model

describes the flow on a Cartesian grid in the upper layer with density ρ of the ocean and

assumes the second layer to be infinitely deep and quiescent. The shallow-water equations

describing the flow in the upper layer are

∂u

∂t+ u ·∇u+ fk × u = −g′∇η + AH∇2u+

τ

ρh− γu|u| (1a)

∂h

∂t+∇ · (hu) = 0 (1b)

where u = (u, v) is the horizontal velocity vector, h the upper-layer thickness, η the interface

displacement (positive downward) and k the unit vector in vertical direction.

The equations include the Coriolis term—the Coriolis parameter f = 2Ω sinφ at latitude

φ with Ω the Earth’s angular velocity—and a dissipation term with horizontal eddy viscosity

AH . The flow is driven by a wind stress τ exerted on the free-surface. A density difference

between the upper and lower layer results in a reduced gravity constant g′. The friction

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between the two fluid layers is modeled by a quadratic stress weighted by the coefficient γ.

At each time the flow is fully described by the state vector X = (u, h).

The specific domain Pierini (2006) used to model the KE is presented in figure 1. A

schematic implementation of the Japanese coastline proved to be essential for capturing the

dynamics of the KE (Pierini 2008). The flow is driven by a constant-in-time wind stress (with

an amplitude of 5× 10−2 N m−2), which is an analytical approximation of the climatological

double-gyre wind field. In our study, the wind-stress field is perturbed by a stochastic wind

field (having an amplitude of 2.3× 10−2 N m−2), which is Gaussian correlated in space with

a correlation length of 2000 km. The amplitude of the stochastic wind field is uncorrelated

in time (white noise). The parameter values can be found in table 1 and further details on

the model formulation and implementation in Pierini (2006).

b. Particle filtering

The shallow-water model can be used to give a forecast for the Kuroshio Extension, when

an initial state X0 is provided. Pierini et al. (2009) showed for realistic parameter values

that the system has a positive Lyapunov exponent. The chaotic nature of the system limits

the prediction time, as initial perturbations grow over time. The uncertainty in the initial

state is assigned to probability density function p(X0). Moreover, the model is not an exact

representation of reality as imperfections are present in the model equations, parameters

and boundary conditions. We model these uncertainties with stochastic perturbations to

the wind stress field. The shallow-water model described in the previous section can be

considered as a discrete time estimation problem. The evolution of the state vector, Xk, is

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described by the following system model

Xk+1 = Mk(Xk, ζk), (2)

where Mk is the system’s propagator and ζk is a zero-mean, white noise sequence.

Observations Yk become available at discrete times. The aim of a data assimilation

technique is to obtain a conditional probability density function of the current state at

t = tk given all available information: p(Xk|Y1:k). Taking a Monte Carlo approach, p(Xk) is

randomly sampled by a weighted ensemble of N model realizations X ik (also called particles),

pN(Xk) =N∑

i=1

wikδ(Xk −X i

k) (3)

with weights wik. The initial state of each ensemble member is uniformly drawn (wi

0 =

1N ) from the initial p.d.f. p(X0). In a particle filtering method pN(Xk|Y1:k) is obtained

recursively in a prediction step and an update step. Assume that pN(Xk−1|Y1:k−1) is known,

i.e. the particle states X ik−1 and the weights wi

k−1 are known. Now, the next observation Yk

becomes available. In the prediction step each particle X ik−1 is integrated forward in time

to obtain the state vector at the new time X ik. The probability pN(Xk|Y1:k−1) follows from

pN(Xk|Y1:k−1) =N∑

i=1

wik−1δ(Xk −X i

k). (4)

In the update step Bayes’ theorem is exploited to get

pN(Xk|Y1:k) =p(Yk|Xk)pN(Xk|Y1:k−1)

p(Yk). (5)

Inserting the p.d.f. (3) for p(Xk|Y1:k) in equation (5) yields for the new weights:

wik =

p(Yk|X ik)

p(Yk)wi

k−1. (6)

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Here, the probability of the observation p(Yk) can be considered as a normalization fac-

tor. This normalization can also be accomplished by demanding that∑N

i=1 wik = 1. The

probability of the observation given the model p(Yk|X ik) is directly linked to the (known)

observational error. For example, for an univariate measurement Yk with a Gaussian dis-

tribution for the measurement error (with standard deviation σobs) this probability follows

from p(Yk|Xk) ∼ exp(−(H(X ik)− Yk)2/2σ2

obs). Here, H(X ik) is the model equivalent of the

observation calculated using the observation operator H.

For an overview of particle filtering in geophysical systems the reader is referred to

Van Leeuwen (2009). The particular particle filter method described here is known as Se-

quential Importance Sampling (SIS) (see e.g. Doucet et al. (2001)). In contrast with other

data assimilation techniques, particle filtering does not require a linearization of the model

around the current state, nor does it assume Gaussian statistics for the state variables. An

advantage of SIS is that assimilating observations changes the weight, but leaves the particle

X ik itself unchanged. Hence, the ensemble can be run beforehand, and the impact of differ-

ent sets of observations can be calculated afterward. A major problem of SIS is that after a

number of observations the weight is concentrated on a small number of particles. The effec-

tive number of particles can be estimated by Neff = 1/∑N

i=1(wik)

2. The traditional solution

of this degeneracy of the ensemble is to resample p(Xk|Y1:k) with an altered set of particles.

A common technique of resampling consists of making copies of particles with a high weight

and discard particles with a low weight. Due to the stochastic forcing and chaotic dynamics

a particle and its copy will diverge over time and become uncorrelated. Resampling, how-

ever, breaks the desired property of evaluating observations as a post-processing step after

the time integration of the ensemble is completed.

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c. Entropy based predictability measures

The application of the particle filter method allows us to sample the non-Gaussian proba-

bility distribution of the state vector. To specify the amount of uncertainty in the probability

density function, entropy based measures are favorable. In this paper we use two measures

that are adopted to quantify the predictability of an ensemble forecast. The first is the

predictive power (PP) introduced by Schneider and Griffies (1999), which is a measure of

the uncertainty relative to the climatological variance. The second is the potential pre-

diction utility (PPU), as introduced by Kleeman (2002), which additionally incorporates a

signal-to-noise component in the measure (see also Xu (2006)).

The predictive power is based on the entropy Sp(X), a measure for the uncertainty associ-

ated with the p.d.f. p(X) of variable X (Shannon 1948). The differential entropy is defined

as

SX ≡ −κ

∫p(X) ln p(X)dX. (7)

Here, κ is a constant which determines the unit of entropy. The particle filtering allows one to

obtain an approximation to the probability density function p(Xk|Y1:k) of the state vector.

With the p.d.f. available, the entropy Sp(X) of the ensemble forecast can be calculated.

Schneider and Griffies (1999) defined the predictive power αX of an ensemble forecast as

αX ≡ 1− exp(−Sq(X) + Sp(X)). (8)

The entropy Sq(X) is calculated from the p.d.f. of the climatology q(X), and it can be consid-

ered as the uncertainty when only the climatological mean is known. Note that differential

entropy (7) is not scaling invariant and cannot be compared directly to the discrete entropy.

Two differential entropies can be compared as long as the reference scales are the same. The

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PP is limited to the range 0 ≤ αX ≤ 1. If the ensemble analysis is equal to the climatological

mean (Sp(X) = Sq(X)), its PP is zero. When the entropy of the ensemble forecast reduces

(for instance by assimilating observations), the predictive power will increase. Note that

due to the exponential form PP is most sensitive in the lower part in the scale, if Sp(X) is

close to Sq(X). Schneider and Griffies (1999) defined the predictive power to measure the

decrease in uncertainty of the full state vector, and set κ = 1/m with m the dimension of

the state vector. This makes it possible to compare entropies of random state vectors of

different dimensions.

The potential prediction utility is based on the relative entropy for p(X),

RX ≡∫

p(X) lnp(X)

q(X)dX, (9)

which is the information gain over the reference p.d.f. q(X). The relative entropy for a

continuous p.d.f. is scaling invariant. For the predictive utility this reference distribution

is the climatological or equilibrium probability distribution. Like the predictive power the

PPU is zero when the probability distribution of the analysis is equal to the climatological

distribution. The PPU of the analysis increases when the spread in probability distribution

decreases, but also when the analysis relates to a relatively rare event. There is no theoretical

upper limit for the PPU. Kleeman and Majda (2005) point out that PPU calculated over

the full state vector RX decreases monotonically over time as uncertainty increases (see also

Cover and Thomas (1991)).

In this paper we opt to calculate the predictive power αs for a single scalar quantity s, e.g.

the kinetic energy E integrated over the KE region, following Pierini (2006). The reasoning

behind determining the predictive power for the kinetic energy is that one is not always

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interested in reducing the uncertainty in the full state vector space. Our design aim is to

follow the path transitions of the Kuroshio Extension and hence we optimize for the kinetic

energy in the KE region. The entropies Sq(s) and Sp(s) are calculated from the univariate

p.d.f. q(s) and p(s) and we set κ = 1 in (7). If p(s) and q(s) are Gaussian probability

distributions with variances σ2p and σ2

q , the predictive power is given by αs = 1 − σ2p/σ

2q .

This formulation (or its multivariate counterpart) are often used in predictability studies

that are based on the analysis error covariance.

d. Set-up of the identical-twin experiment

Instead of forecasting the actual Kuroshio Extension by assimilating real observations in

the shallow-water model, an identical-twin experiment will be performed. Here, one model

realization is considered to be the ‘true’ evolution of the KE. Equivalent observations are

produced by taking measurements from this synthetic truth and adding observational errors.

This allows us to produce observations for different variables, like the SSH or the velocity

field, with different error distributions. Then, an analysis is performed by assimilating

the observations in an ensemble run of the shallow-water model using the particle-filtering

technique.

The advantage of the identical-twin experiment is that we are sure that the evolution

of the ‘true’ KE is captured by the ensemble run with the shallow-water model. A failure

to describe the truth is then not caused by the model, but is either caused by insufficient

observations or the inadequacy of the data assimilation method.

In this study we use a particular set-up for the ensemble: At each time the ensemble

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samples the climatological probability density function q(X). This is achieved by drawing

the initial conditions X i0 from the climatological distribution. In practice, we obtained the

initial states by taking snapshots, two years apart, of the state vector from a long model run.

For each particle (N = 512) the shallow-water model driven by the stochastic wind stress is

integrated over 40 years.

Assimilating the synthetic observations using the particle filter (6) will change the weight

of the ensembles members. Starting with a uniform distribution of the weights (wi0 = 1/N)

the initial probability distribution is equal to the climatology probability distribution. This

initial state has zero predictive power and zero potential prediction utility. Hence, any

increase of the PP of the ensemble is due to the information the observations provide.

3. The impact of observations on the predictability

a. The predictive power of an ensemble forecast

As an illustrative example we first use the predictive power and potential prediction

utility to determine the predictability time of an ensemble prediction with the shallow-water

model under stochastic wind forcing. Essentially, this is a predictability study of the second

kind (Lorenz 1975), where the influence of uncertain boundary conditions — in our case

in the wind stress — on the predictability is determined. For this purpose we have run

a 96-member ensemble that starts from identical initial conditions. The initial state X i0

is obtained from a 50-year spin-up model run. With no uncertainty present in the initial

conditions (i.e. the differential entropy is infinite) the ensemble starts with a predictive

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power equal to unity. This is similar to the set-up used by Griffies and Bryan (1997) to

study the predictability of the North Atlantic multidecadal climate variability. Their study

then motivated Schneider and Griffies (1999) to introduce the predictive power as a measure

for the uncertainty in ensemble model forecast.

In figure 2a the time evolution of the kinetic energy of the Kuroshio Extension is plotted.

This kinetic energy is defined by E = 12

∫A |u|2dS, where A is the region of the free jet as

defined in fig. 1. The energy signal clearly reveals the decadal transitions of the KE between

a low energy and high energy state. The two KE states are less clear from time series of

potential vorticity or enstrophy integrated over the same region. With knowledge of the

kinetic energy one could already provide a good estimate for the path length and the mean

latitude of the jet (Pierini et al. 2009). The first strong divergence of the ensemble occurs

after 5 years, when the transition from the low energy state to the high energy state starts.

Then, after 8 years the signals branch. The main branch relates to a fast transition to the

low energy state, while the other branch reveals a slow decay of the kinetic energy for two,

four or six years. As the signals become desynchronized the predictability of the system is

limited to a certain time period.

In figure 2b the predictive power of the ensemble is given for a time span of 40 years.

Here, the PP (8) is calculated using p(E) instead of the probability density function p(X)

of the full state vector. This indicates that we investigate the PP of the energy signal and

not of the complete information the forecast provides. An estimate for the climatology p.d.f.

q(E) is obtained by binning the time series of the N = 512 particle ensemble (described in

the previous section) over 23 bins. The approximation for p(E) is obtained over the same

bins. Note that the maximum PP is limited by the bin spacing ∆E as the minimum entropy

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Sp(E) = ln∆E (Cover and Thomas 1991), i.e. when all the particles fall within one bin.

During the first 5 years the predictive power in figure 2b remains high. Then, over a one-

year period the predictive power drops to αE = 0.5. Subsequently, the PP rises and ranges

between αE = 0.6 and 0.8 for years 6 – 13. This rise of αE is the result of the energy plateaus

in the signal. This causes a smaller uncertainty in E, while the signals clearly become

desynchronized. In this period the weighted ensemble p.d.f. for E is strongly bimodal or

trimodal. Schneider and Griffies (1999) defined the predictability life span as the lead time for

which the 95% confidence interval of the predictive power does not include zero. This requires

detailed information on the error in the prediction probability distribution p(E), which is

not available from one ensemble. In reality an ensemble prediction loses its usefulness at an

earlier time, as there are more strict requirements for the forecast uncertainty. For simplicity

we define the predictability life span of the ensemble as the time αE drops below 0.25. This

definition results in a predictability life of 13 years. During this time it is possible to predict

whether the KE is in the high-energy state or in one of the other states.

b. Measuring the impact of observations

Now, we return to our objective of measuring the impact of observations. For this purpose

we solely use the 512 particle ensemble which samples the climatology distribution at any

time. The energy evolution for all the particles is given in figure 3. The unweighted mean

of this ensemble is always equal to the climatological mean field, and its predictive power

is equal to zero. Clearly, the climatological p.d.f. is bimodal. As the analysis starts from

the climatological distribution, a data assimilation method that can handle non-Gaussian

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statistics is required.

One particle is selected and is considered to be the true behavior of the Kuroshio Exten-

sion (e.g. the white curve in figure 3). From this synthetic truth almost any observation can

be produced, as we have access to the full state vector. For now, we opt to limit our studies

to four different locations or ‘moorings’ where observations of both velocity components are

obtained. The velocity components are directly available from the shallow-water model. For

this set-up the observation operator H(X ik) simply selects the velocity components from the

state vector at the mooring location.

The selection of the four locations is based on the climatological mean and variance of

the SSH (see figure 4). In practice sea-surface height fields are readily available from satellite

observations. The moorings are located in the circulation cell to the south of Japan (A) and

at the positions where the local maxima of SSH variance occur (B, C & D). Here, mooring

B is located at the absolute maxima, mooring C is in the KE near the first meander, while

mooring D is at a more eastern location in the KE region. The synthetic velocity observations

for the four moorings are given in figure 5. Observations for the zonal and meridional

velocity components are produced at a monthly data rate (each 28 days) and have a Gaussian

measurement error with a standard deviation of σobs = 0.1m/s. Observation errors for the

zonal and meridional velocity are not correlated, as such they can be assimilated sequentially.

With the particle filtering technique (SIS) described in section 2, the observations are

used to improve our ensemble forecast. Cycling through the observations in a sequential

order, the weights of each particle are adjusted according to equation (6). The resulting

analysis is the weighted ensemble of all the particles. The probability density function p(E)

of the analysis is given in figure 6. It becomes immediately clear that observations from

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moorings B are most effective in reducing the uncertainty. With observations from mooring

C we can not determine whether the KE is in the high or low energy state for the first

three years. Assimilating data from mooring A rightly puts the system in the high energy

state, but with a large uncertainty in the initial years. If the first KE transition occurs

results improve for moorings A and C. The analysis using data from mooring D is the worst.

During the first ten years we are unable to determine the state of the KE from the analysis.

For times longer than three years the ensemble becomes degenerated for all cases. Then

the analysis does not capture the true behavior nor does it give an accurate representation

of the probability distribution. The observations at location B have a high signal-to-noise

ratio. The first few observations will rapidly reduce the uncertainty from its initial value.

As a consequence, a large number of particles is discarded after each observation and the

ensemble rapidly degenerates. In figure 7 the effective number of particles Neff is given for

the four moorings. For mooring B all the weight effectively collapses to a few particles after

two years. The analysis still follows the first transitions of the KE, due to the slow divergence

of particle trajectories — the predictability life is 13 years. One particle close to the truth

will follow the true evolution for a few years. A degenerated ensemble does not accurately

capture the real uncertainty, and it is not guaranteed that it can track the true evolution.

For instance, the degenerated analysis for mooring B fails to describe the KE transitions

between year 12 to 20.

At each time a discrete probability distribution for the kinetic energy can be obtained by

binning the weighted ensemble. The number of bins used to obtain the histogram of p(E)

and q(E) is√N . The probability density pj(E) is the sum over the weights from particles

that fall in bin j divided by the bin size ∆E. An estimate is obtained for Sp(E) (with

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κ = 1) by numerical integration over the discrete histogram, i.e. 1κ

∑j pj(E) log pj(E)∆E.

The unweighted ensemble gives an estimate for the climatology p.d.f. of Sq(E) = −0.45.

The predictive power αE is given in figure 7. The graph nicely shows that assimilating

observations from mooring B raises the predictive power of the ensemble rapidly from zero

to its limiting value after three years. Note that reaching the limiting value for PP means

that all the particle weights fall within one bin. Observations from the other mooring raise

the predictive power at a slower rate. For mooring C and D the sudden increase in predictive

power is related to the transition from the high energy to the low energy state. This gives

a clear signal in the velocity at these locations.

While a single analysis gives already much insight into the impact of different observa-

tions, a number of problems arise. The selected synthetic truth might show some sporadic,

unrepresentative behavior. Moreover, this synthetic truth starts from a state which might

be more or less sensitive to perturbations. This can produce a different initial increase of

the predictive power. Picking another particle to act as the synthetic truth, creating new

observations and performing a new analysis will give different results. We exploit this by

selecting each of the 512 model realizations to be the synthetic truth at a time. This leads

to 512 different evolutions of the entropy Sp(E). The average entropy Sp(E) is then used to

define the predictive power αE = 1− exp(−Sq(E)+Sp(E)). In figure 8 the predictive power is

given up to twenty years. By assimilating data from mooring B the predictive power rapidly

increases to a value of 0.7 in one year. For mooring C the same αE is reached after two

years, while five years are required when assimilating data from mooring A or D.

In the special context of sequential importance sampling and the specific discrete approx-

imations (3) for p(X) and q(X) there is an accurate way of calculating the PPU for the full

17

state vector. For an ensemble where the initial states are drawn from q(X), the discrete ap-

proximation is qN(X0) =∑N

i=1 wi0δ(X0−X i

0) with wi0 = 1/N . If this ensemble is integrated

forward in time it still represents the climatological p.d.f, i.e. qN(Xk) =∑N

i=1 wi0δ(Xk−X i

k)

is also an approximation of q(X). Assimilating k observations changes the weights and an

approximation for p(Xk|Y1:k) is given by pN(Xk|Y1:k) =∑N

i=1 wikδ(Xk − X i

k). Inserting

these approximations in (9) yields

RX ≈N∑

i=1

wik ln

wik

wio

. (10)

Note that this equation is only valid at the time an observation is made. It does not capture

the decrease of RX , which occurs between observations. The practical limit for the relative

entropy with an ensemble size N is RX = lnN , which we can use to normalize the PPU.

The normalized PPU averaged over 512 realizations of the synthetic truth is presented in

figure 8b. The behavior of RE is similar to the αE. The differences between the two measures

mainly originate from the different sensitivity along their range. Due to the exponential term

in the PP, its sensitivity is larger in the lower part of the scale, while the PPU is more sensitive

in the higher region of the scale. These results validate the use of αE to measure the impact

of observations.

The degeneracy of the ensemble weights and the numerical limit to the predictive power

essentially invalidate any results on predictive power after three years. The sampling of the

probability distribution is too poor to obtain accurate values for the entropy. Increasing the

numbers of particles could alleviate both problems. A more realistic option is to restrict the

analysis to a time span of one year. This essentially yields the increase of predictive power

after assimilating observations over a one-year period. For this approach the ensemble time

18

span can be reduced from 40 years to one year. In the following sections we split the original

ensemble (N = 512) in to a single synthetic truth and in 20 440 one-year segments. This

larger ensemble can be used for an uncertainty analysis over a one year period. Choosing the

approach does not allow to perform data assimilation over a number of decades, which would

give an analysis of several Kuroshio Extension path transitions. The one-year analysis can

be used to target the specific states of the KE and determine which observation locations

decrease the uncertainty in the analysis.

4. Optimal monitoring of the Kuroshio Extension

a. The spatial decrease of uncertainty in the potential vorticity

The energy of the jet E is a spatially integrated signal, and hence the analysis does

not give any information where the uncertainty is decreased by the assimilation of the ob-

servations. Hence, we take another approach by investigating how the uncertainty in the

potential vorticity (PV) field Q(x, t) = (ζ + f)/h is decreased by assimilating observations.

Here, ζ is the vertical component of the relative vorticity, ∇ × u. Potential vorticity is a

conservative advectice tracer only influenced by forcing and dissipation. Distributions of PV

contain nearly all the information about the flow dynamics.

To investigate the uncertainty in the PV the predictive power is defined as

αQ(x, t) = 1− exp(−Sq(Q)(x, t) + Sp(Q)(x, t)) (11)

The climatological entropy of the PV at point x, Sq(Q)(x, t), is calculated using the Qi(x, t)

from the unweighted ensemble members. The climatology mean and variance of the PV are

19

given in figure 4. Observations within a time span of one year are assimilated, yielding a

reduction of the entropy Sp(Q)(x, t) in the weighted ensemble. This result is dependent on

the time period for which observations are taken. Hence, three different time ranges, starting

after 4, 8 and 12 years are considered (for reference, see figure 6). The results of the analysis

are presented in figure 9.

The previous section revealed that, on average, velocity observations taken at locations

B and C are successful for reconstructing the kinetic energy of the Kuroshio Extension.

The study of αQ(x, t) reveals that there are differences in the performance of the moorings

depending on the state of the KE. For mooring B we obtain good results for years 8 and

12. In year 8 the KE is in the high-energy state, while during year 12 a transition from

the high- to low-energy state occurs. The analysis is particularly good for the first meander

of the KE, also indicated by the high PP in these regions. Note that the PP is large in

quiescent regions with small PV gradients. Regions where PV is more variable, like for the

eastern meandering part of the KE or where the Kuroshio boundary current detaches from

the Japanese coastline, are less predictable. The eastern part of the Kuroshio Extension is

actually stronger influenced by the stochastic wind forcing. Thus part of the variability here

is not correlated with the state of the KE. When the KE has just switched to the low energy

state (year 4) the recirculation gyre is well retrieved but the reconstruction of the KE path

is less successful. Low values for PP are thus either due to large short-time variability of the

KE in the current state or due to dynamics that are not correlated to the observations. On

the other hand a region with high PP is correlated with the observations and the analysis

has a small uncertainty.

The analysis for year 4 which uses observations from mooring C is more successful in

20

retrieving the first meander of the KE. Note that the analysis for mooring B actually captures

the detachment of the Kuroshio boundary current and its path quite good upstream of

mooring C. This indicates that information from mooring C is essential for retrieving the

path of the KE in the meandering, low-energy phase. During the elongated-jet, high-energy

state of the KE (year 8) does not give good results as the mooring C is located south of

the KE path in a quiescent region. The success in reconstructing the eastern KE for the

transition in year 12 is a bit surprising, as the first meander itself is not accurately recovered.

The analysis using data taken at location D yields bad results for all the stages of the

KE decadal oscillation. The poor results are mainly due to the bad signal-to-noise ratio of

the observed velocity signal. Data from mooring A yield bad results overall, but leads to a

good reconstruction of PV during the elongated-jet state of the KE (year 8). In this case

the southern recirculation gyre is stronger than in the other periods. In these other stages

the variation in the observed signal at mooring A is below the noise level.

b. The optimal measurement location

The four mooring locations chosen were based on the variance of the SSH field for a

long climatological run. Assimilating the velocity data from these different locations yielded

varying results for reconstructing the path and kinetic energy of the KE. Mooring B yielded

good results for most phases of the decadal oscillation, but its resolution of the eastern part

of the KE path is somewhat lacking. Overall mooring C gives good reconstruction of the

KE path, but completely fails to capture the position of the recirculation gyre and first KE

meander during the high-energy state. The opposite is true for the analysis with observations

21

from mooring A bad results overall except for the elongated-jet state. Assimilating data over

a longer period favors data from mooring B or C (figure 8). Most likely none of these mooring

positions is the optimal location. What is the single optimal location for reconstructing the

decadal oscillations in the kinetic energy of the KE? And is this location also optimal when

sea-surface height observations are assimilated instead of velocities?

Our answer to this question is presented in figure 10. Instead of producing observations

for the four locations, we apply our method to each and every grid point. Assimilating the

observations for a given point yields a predictive power of αE (averaged over 40 synthetic

truths). In figure 10 this value is mapped to the location where the observations originate.

The optimal measurement location is simply the one tagged by the highest predictive power.

For an analysis based on velocity observations the optimal location is at (139E,33N).

Instead of using observations of the two velocity components we can opt to assimilate

sea-surface height. As we now assimilate only one variable the observation frequency was

doubled (biweekly) to obtain the same number of observations. The observational error

added to the synthetic truth is 10 cm/s. These settings result for mooring A,C and D in

an average increase of PP with a similar rate as found for velocity observations (figure 8).

Assimilating velocity observations from location B increases the PP faster than using sea-

surface height. Another difference is that using SSH observations (instead of velocities) from

mooring C it accurately reconstructs the PV field for year 8, while it fails for year 12. When

assimilating SSH observations the optimal location is at (139.5E,32N). For both velocity

and sea-surface height observations, the optimal location is located in the region where the

Kuroshio boundary current detaches from the coast and has a large meander.

The question remains if the optimal observation location is optimal for both the high-

22

energy elongated state of the KE and for low-energy contracted state. To check this, we

selected the synthetic truths that correspond to one of the KE states and calculated the

average PP. The result is presented in figure 11 for both the low- and high energy state.

For the low-energy state the optimal location coincides with the ‘overall’ optimal location

determined from figure 10. However, during the high-energy state the optimal location is at

(145.5E,36N) at the point where the energetic jet starts to meander. Measuring velocity

observations in the detached Kuroshio or in the first meander of the Kuroshio Extension gives

good predictive power. Note that measuring SSH along the KE path during the elongated

jet state does not give good results (figure 10).

Note that the optimal locations for both phases coincide with regions of strong potential

vorticity gradients (see figure ??). These regions are susceptible to barotropic instability,

and hence regions where strong growth of perturbations can be expected. This also confirms

with our finding that only a small increase of αQ(x, t) is obtained in regions with large PV

gradients. For the high-energy elongated state the largest PV gradients are south of Japan

where the Kuroshio current first separates and in the first meander of the KE. Pierini (2006)

argued that during this state the recirculation gyre and the KE are virtually isolated dynam-

ically by the presence of the strong cyclonic meander. This view is supported by our findings

that the optimal location is then in the KE itself (figure 11), and that observations from

mooring B and C do not yield a significant decrease uncertainty in the southern circulation

gyre (figure 9). Note, that after the cyclonic meander the Kuroshio current reattaches to the

coast, where lateral friction provides a source of cyclonic PV. This is not the case in the low-

energy contracted state, where the Kuroshio remains a free jet (KE) after it detaches from

the coast. Hence, there is a stronger coupling between the KE and the southern circulation

23

gyre.

5. Summary and conclusions

A general method is proposed to find which observation is most effective in decreasing

the uncertainty in an ensemble model forecast of the Kuroshio Extension chaotic dynamics

produced by a reduced-gravity shallow-water model. At all times the unweighted ensemble

represents the climatological mean and the probability distribution. As such, the ensemble

itself has no predictive skill. Using a particle filter approach the weight of each ensemble

member (particle) is changed according to the likelihood of the observation given the current

particle state. Using the basic sequential importance sampling technique, assimilating obser-

vations is a post-processing step. As a result, the impact of a variety of observations on the

forecast uncertainty can be investigated. Assimilating a series of observations changes the

uncertainty in the weighted ensemble. The decrease in uncertainty relative to the change-

ability of the climatology is measured using the predictive power. The search for an optimal

observational strategy is then quantified in terms of predictive power.

On average, each subsequent observation reduces the uncertainty in the weighted en-

semble. In the long term the uncertainty converges to a value determined by the accuracy

and effectiveness of the observation, and by the divergence rate of the model state. The

desired accuracy for sampling the long term probability distribution prescribes the required

ensemble size. In practice, the required number of model integrations is not feasible and

all the weight becomes concentrated on a few particles. The traditional resolution for the

degeneracy of the ensemble — resampling — would require a separate ensemble run for each

24

set of observations. Instead, we limit the analysis of the predictive power to the short term,

i.e. before the ensemble degenerates.

This methodology is applied to find the optimal location to follow the path transitions of

the KE in the context of a reduced-gravity model. Specifically, an identical-twin experiment

is used, where the ‘true’ evolution of the KE is synthetically produced by the numerical

model. The kinetic energy of the KE is a good proxy to follow the chaotic decadal transitions

of the KE. We have looked for the observation location that allows for the reconstruction of

the kinetic energy with the lowest uncertainty, as measured by the predictive power. The

optimal location for following the KE is south of the Boso Peninsula. Here, the Kuroshio

typically has a large meander before entering the Pacific as a free jet. The shift of this

meander and the recirculation gyre along the Japanese coastline is a clear indicator of the

state of the KE. At a good measurement location the signal-to-noise ratio has to be good

and there are no elongated periods where the signal is flat. These criteria are satisfied in a

region where strong PV gradients are continuously shifting.

In our study observations of the velocity components or sea-surface height at a single

location are assimilated, yielding an increase of predictive power defined for a univariate

quantity (the kinetic energy of the KE). The applicability of the methodology is however

much broader. Particle filtering allows for the simultaneous assimilating multivariate ob-

servations (like SSH and velocities) at multiple locations. We have seen that the success of

observations in improving the forecast depends on the current state of the KE. Combining

observations that are successful during different states lead to an even better observational

system. Predictive power is an entropy based measure equally suitable for measuring the

uncertainty in multivariate probability distributions. This allows us to optimize the obser-

25

vational strategy for multiple conditions, say, the kinetic energy of the jet and the strength

of the recirculation gyre.

Optimal observation locations correspond to regions with strong potential vorticity gra-

dients. Strong PV gradients are a condition for barotropic instabilities to occur, which can

lead to fast growth of perturbations. Measuring at these location would prevent this poten-

tial growth of uncertainty. During the contracted state of the KE the strongest PV gradient

is located south of Japan, where the Kuroshio detaches from the coast. The optimal loca-

tion is in the elongated jet during the high-energy state of the KE. This seems to confirm

with assumption that during this state the KE is dynamically decoupled from the southern

circulation gyre Pierini (2006).

The regional experiments of the Kuroshio Extension (WESTPAC, KERE and KESS)

were situated at different locations (figure 10). In the context of our reduced-gravity model,

the WESTPAC study at 152E would be situated too far east to adequately capture the

transitions of the KE. The variability related to the KE path transitions is here simply too

small to yield a good signal-to-noise ratio. In reality this region is characterized by strong

variability which is caused by baroclinic instabilities. Close to Japan, where the KERE and

KESS are situated, the KE is dominated by barotropic and equivalent barotropic dynamics.

The KESS study captures the dynamics of the first large meander of the KE, a region

with strong PV gradients. As such, observations from KESS should capture the decadal

KE transitions. For the elongated state the path of the KE can be discerned in KERE

observations. As the path is there quite stable, it would be difficult to predict it when the

transition to the contracted state occurs. Our study suggests that it is also worthwhile to

monitor the detachment of the Kuroshio to determine the decadal transitions of the KE.

26

Acknowledgments.

One of the authors (W. K.) is sponsored by the NSO User Support Programme under

grant ALW-GO-AO/08-14, with financial support from the Netherlands Organisation for

Scientific Research (NWO). This work was sponsored by the National Computing Facilities

Foundation (NCF) for the use of supercomputer facilities, with financial support from NWO.

P.J.v.L. is partly sponsored by the National Centre for Earth Observation (NCEO), funded

by the National Environment Research Council (NERC).

27

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31

List of Tables

1 Settings for the reduced-gravity shallow-water model. The equations are

solved numerically on a Cartesian grid with spacing (∆x,∆y) and time step

∆t. The upper layer has a mean depth H and density ρ. The friction between

the upper and lower layer is quadratic with a friction coefficient γ. 33

32

Table 1: Settings for the reduced-gravity shallow-water model. The equations are solvednumerically on a Cartesian grid with spacing (∆x,∆y) and time step ∆t. The upper layerhas a mean depth H and density ρ. The friction between the upper and lower layer isquadratic with a friction coefficient γ.

∆x 20 km∆y 20 km∆t 20minAH 220m2 s−1

H 500mρ 1023.5 kg m−3

g′ 4.41× 10−3 m s−2

γ 5.0× 10−4 m−1

33

List of Figures

1 The computational domain zonally spans 10 700 km from 122E to approxi-

mately 120W at 35N and ranges from 5S to 55N (Pierini 2006). At the

western side an schematic implementation of the Japanese coastline is used.

The climatological wind-stress curl is given by the contour map with negative

values in the light gray area (contour spacing is 1.5× 10−8 N m−3. 37

2 Ensemble forecast for the Kuroshio Extension with a stochastic forced shallow-

water model starting from a single initial state. (a) Evolution of the kinetic

energy E of the KE for the 96 ensemble realizations and (b) predictive power

αE for the kinetic energy signal of the forecast. 38

3 Data from the ensemble run for the identical-twin experiment. The gray lines

indicate the kinetic energy E for the 511 particles, while the white curve

corresponds to the synthetic truth. In the right panel the climatological p.d.f.

q(E) is given. 39

4 Climatological mean SSH field (a) and potential vorticity (b) calculated from

the ensemble. The gray scales represent the standard deviation with re-

spect to the climatological mean. Contour spacing is 5 cm for the SSH and

10−8 s−1 m−1 for the potential vorticity. The thick contours give the 10 cm

SSH level and the 1.6× 10−7 s−1 m−1 PV level. Observations are produced

for the positions labeled A, B, C and D. 40

34

5 Observations (+) for the identical-twin experiment are produced by adding a

Gaussian measurement error to the synthetic truth (drawn). Observations for

u and v are produced for the four locations A, B, C and D as shown in figure 4. 41

6 Analysis of the kinetic energy E of the Kuroshio Extension resulting from

assimilating velocity observations from either location A, B, C or D. The gray

scale plot gives p(E) of the ensemble analysis, with the darkest gray related

to highest probability density. The black line corresponds with the synthetic

truth. 42

7 Predictive power αE (black line) for the analysis (given in figure 6) resulting

from assimilating velocity observations from either location A, B, C or D. The

limit on the predictive power due to the used bin size is depicted by the dotted

line. The effective number of particles Neff is denoted by the gray curve. 43

8 Average predictive power αE (a) and average potential prediction utility (b)

for the analysis calculated by repeating the algorithm with other particles

being selected as the synthetic truth. The quantity is given for the four

locations: A (asterisk), B(plus), C(circle) or D(diamond). 44

9 One-year increase in the predictive power αQ(x, t). Velocity observations from

either mooring A, B, C and D (left to right) are assimilated during one year,

starting at year 4, 8 and 12 (top to bottom). Red contour lines depict the

PV mean from the analysis. Black lines give the PV field from the synthetic

truth. For each analysis the effective number of particles is given in the top

left corner. 45

35

10 A color map of the one-year increase of αE when a) velocity observations or b)

sea-surface height observations are assimilated for the given location. The star

gives the optimal observation location and the plus signs give the locations

of the synthetic moorings A, B, C and D. The diamonds give the position of

the current meter moorings used in KERE (Hallock and Teague 1995), KESS

(Donohue et al. 2008) and WESTPAC (Schmitz Jr. et al. 1982). 47

11 A color map of the one-year increase of αE when velocity observations are

assimilated during a) the low-energy contracted state or b) the high-energy

jet state for the given location. For each synthetic truth a single potential-

vorticity contour is used as a proxy for the Kuroshio Extension trajectory.

The star gives the optimal observation location and the plus signs give the

locations of the synthetic moorings A, B, C and D. 48

36

!"#$%&! !"'$

%&! !"($

%)! !"'$

%)!

!!!$%!!

!!"*%+!

!!,$%+!

!!#*%+!

!-

Figure 1: The computational domain zonally spans 10 700 km from 122E to approximately120W at 35N and ranges from 5S to 55N (Pierini 2006). At the western side an schematicimplementation of the Japanese coastline is used. The climatological wind-stress curl isgiven by the contour map with negative values in the light gray area (contour spacing is1.5× 10−8 N m−3.

37

a) kinetic energy of all ensemble members

0 5 10 15 20 25 30 35 40

0.4

0.6

0.8

1

1.2

time [yr]

E

b) predictive power

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

time [yr]

αE

Figure 2: Ensemble forecast for the Kuroshio Extension with a stochastic forced shallow-water model starting from a single initial state. (a) Evolution of the kinetic energy E ofthe KE for the 96 ensemble realizations and (b) predictive power αE for the kinetic energysignal of the forecast.

38

0 5 10 15 20 25 30 35 40

0.4

0.6

0.8

1

1.2

time [yr]

E

0 2 4

0.4

0.6

0.8

1

1.2

q(E)

Figure 3: Data from the ensemble run for the identical-twin experiment. The gray linesindicate the kinetic energy E for the 511 particles, while the white curve corresponds to thesynthetic truth. In the right panel the climatological p.d.f. q(E) is given.

39

a) sea-surface height

130oE 140

oE 150

oE 160

oE

27oN

30oN

33oN

36oN

39oN

A B

C

D

b) potential vorticity

130oE 140

oE 150

oE 160

oE

27oN

30oN

33oN

36oN

39oN

A B

C

D

Figure 4: Climatological mean SSH field (a) and potential vorticity (b) calculated from theensemble. The gray scales represent the standard deviation with respect to the climatologicalmean. Contour spacing is 5 cm for the SSH and 10−8 s−1 m−1 for the potential vorticity. Thethick contours give the 10 cm SSH level and the 1.6× 10−7 s−1 m−1 PV level. Observationsare produced for the positions labeled A, B, C and D.

40

0 10 20 30 40

0

0.5

u [

m/s

]

−0.5

0

0.5

v [m

/s] B

0 10 20 30 40

0

0.5

−0.5

0

0.5 C

0 10 20 30 40

0

0.5

time [yr]

u [

m/s

]

−0.5

0

0.5

v [m

/s] A

0 10 20 30 40

0

0.5

time [yr]

−0.5

0

0.5 D

Figure 5: Observations (+) for the identical-twin experiment are produced by adding aGaussian measurement error to the synthetic truth (drawn). Observations for u and v areproduced for the four locations A, B, C and D as shown in figure 4.

41

E

B

0 10 20 30 40

0.4

0.6

0.8

1

1.2 C

0 10 20 30 40

0.4

0.6

0.8

1

1.2

time [yr]

E

A

0 10 20 30 40

0.4

0.6

0.8

1

1.2

time [yr]

D

0 10 20 30 40

0.4

0.6

0.8

1

1.2

Figure 6: Analysis of the kinetic energy E of the Kuroshio Extension resulting from assimi-lating velocity observations from either location A, B, C or D. The gray scale plot gives p(E)of the ensemble analysis, with the darkest gray related to highest probability density. Theblack line corresponds with the synthetic truth.

42

0 10 20 30 400

0.5

1α

E

B

0 10 20 30 4010

0

101

102

0 10 20 30 400

0.5

1C

0 10 20 30 4010

0

101

102

Neff

0 10 20 30 400

0.5

1

time [yr]

αE

A

0 10 20 30 40

101

102

0 10 20 30 400

0.5

1

time [yr]

D

0 10 20 30 40

101

102

Neff

Figure 7: Predictive power αE (black line) for the analysis (given in figure 6) resultingfrom assimilating velocity observations from either location A, B, C or D. The limit on thepredictive power due to the used bin size is depicted by the dotted line. The effective numberof particles Neff is denoted by the gray curve.

43

a) predictive power for E

0 5 10 15 200

0.2

0.4

0.6

0.8

1

time [yr]

αE

b) average potential prediction utility for X

0 5 10 15 200

0.2

0.4

0.6

0.8

1

time [yr]

RX/ln

N

Figure 8: Average predictive power αE (a) and average potential prediction utility (b) forthe analysis calculated by repeating the algorithm with other particles being selected as thesynthetic truth. The quantity is given for the four locations: A (asterisk), B(plus), C(circle)or D(diamond).

44

mooring A mooring B

! !"# !"$ !"% !"& !"' !"( !") !"* !"+ #

Figure 9: One-year increase in the predictive power αQ(x, t). Velocity observations fromeither mooring A, B, C and D (left to right) are assimilated during one year, starting at year4, 8 and 12 (top to bottom). Red contour lines depict the PV mean from the analysis. Blacklines give the PV field from the synthetic truth. For each analysis the effective number ofparticles is given in the top left corner.

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mooring C mooring D

! !"# !"$ !"% !"& !"' !"( !") !"* !"+ #

Figure 9: (continued) One-year increase in the predictive power αQ(x, t). Velocity obser-vations from either mooring A, B, C and D (left to right) are assimilated during one year,starting at year 4, 8 and 12 (top to bottom). Red contour lines depict the PV mean fromthe analysis. Black lines give the PV field from the synthetic truth. For each analysis theeffective number of particles is given in the top left corner.

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a) the optimal spot for velocity observations

b) the optimal spot for sea-surface height observations

Figure 10: A color map of the one-year increase of αE when a) velocity observations or b)sea-surface height observations are assimilated for the given location. The star gives theoptimal observation location and the plus signs give the locations of the synthetic mooringsA, B, C and D. The diamonds give the position of the current meter moorings used in KERE(Hallock and Teague 1995), KESS (Donohue et al. 2008) and WESTPAC (Schmitz Jr. et al.1982).

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a) the optimal spot during the contracted Kuroshio Extension state

b) the optimal spot during the elongated Kuroshio Extension state

Figure 11: A color map of the one-year increase of αE when velocity observations are assim-ilated during a) the low-energy contracted state or b) the high-energy jet state for the givenlocation. For each synthetic truth a single potential-vorticity contour is used as a proxy forthe Kuroshio Extension trajectory. The star gives the optimal observation location and theplus signs give the locations of the synthetic moorings A, B, C and D.

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